The existence of \(g^{-1}\)  is guaranteed by the Whitney immersion theorem red[1]}, which states that for any manifold of dimension \(m\)  there exists a one-to-one mapping in a Euclidean space of dimension at least \(2m\)  (\(2m-1\)  if the manifold is smooth).
In addition, by Nash-Kuiper embedding theorem red[2]} this can be done isometrically, i.e. without any metric distortion, which addresses the last point of the definition.
